40 2. Background

it is negative another equation for the same plane may be obtained

by multiplying through by −1. It is not without loss of generality to

assume the coeﬃcient of x is 1, since the plane defined by y + z = 0,

for example, has no equation with an x-coeﬃcient of 1.

Exercise 2.6.1. Here are some proofs you can try that don’t involve

induction:

(i) ¬(∀m)(∀n)(3m + 5n = 12) (over N)

(ii) For any integer n, the number n2 + n + 1 is odd.

(iii) If every even natural number greater than 2 is the sum of two

primes, then every odd natural number greater than 5 is the

sum of three primes.

(iv) For nonempty sets A and B, A × B = B × A if and only if

A = B.

(v)

√

2 is irrational. (Hint: work by contradiction, assuming there

is a fraction of integers in least terms that equals

√

2.)